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<p align="center"><font size="4"><b>Classification of thunderstorm and non-thunderstorm 
  days in<br>
  Calcutta (India) on the basis of linear discriminant analysis</b></font></p>
<p>&nbsp;</p>
<p align="center"> <font size="2">S. GHOSH</font><br>
  <i>Atmospheric Science Research Group, Environmental Science Programme<br>
  Faculty of Science, Jadavpur University, Calcutta 700032</i><br>
  <font size="2">P. K. SEN</font><br>
  <i>Department of Mathematics, Jadavpur University, Calcutta 700032</i><br>
  <font size="2">U. K. DE</font><br>
  <i>Atmospheric Science Research Group, Environmental Science Programme<br>
  Faculty of Science, Jadavpur University, Calcutta 700032</i></p>
<p align="center"> Received June 6, 2002; accepted August 26, 2003</p>
<p align="center">&nbsp;</p>
<p align="center">RESUMEN</p>
<p align="justify">En el presente trabajo se aplicaron dos t&eacute;cnicas multivariadas 
  sofisticadas, el &#8220;An&aacute;lisis de componente principal&#8221; y el 
  &#8220;An&aacute;lisis lineal discriminante de dos grupos&#8221;, para analizar 
  el clima pre-monz&oacute;nico de Calcuta (India) y predecir las tormentas pre-monz&oacute;nicas 
  de ese lugar. El trabajo se desarroll&oacute; en las siguientes etapas: i) An&aacute;lisis 
  con 20 par&aacute;metros din&aacute;micos y termodin&aacute;micos derivados 
  de los datos diarios de la radiosonda de Calcuta, ii) An&aacute;lisis con 10 
  par&aacute;metros nuevos que son los 10 componentes principales formados con 
  los 20 par&aacute;metros originales. El estudio indica que se puede construir 
  un &iacute;ndice conocido como &#8220;Funci&oacute;n lineal discriminante&#8221; 
  (FLD), para predecir el clima pre-monz&oacute;nico de Calcuta. Adem&aacute;s, 
  la investigaci&oacute;n revela que si se reducen las dimensionalidades de las 
  matrices de datos, la exactitud de los resultados se mejora.</p>
<p align="center"><br>
  ABSTRACT</p>
<p align="justify">The two sophisticated applied multivariate techniques, &#8216;Principal 
  Component Analysis&#8217; and &#8216;Two-group Linear Discriminant Analysis&#8217; 
  have been applied in the present work to analyze the pre-monsoon weather in 
  Calcutta (India) and hence to forecast the pre-monsoon thunderstorms there. 
  The work has been performed in the following two stages: i) Analysis with 20 
  thermodynamic and dynamic parameters derived from daily radiosonde data in Calcutta; 
  ii) Analysis with 10 newly formed parameters which are actually the first 10 
  principal components formed with the 20 original parameters. The study indicates 
  that an index known as &#8216;Linear Discriminant Function&#8217; (LDF) may 
  be constructed to predict the pre-monsoon weather in Calcutta. Not only that, 
  the study also reveals that if the dimensionalities of the data matrices are 
  reduced, then the accuracy of the results may improve.</p>
<p align="justify"> <b>Key words: </b>Equivalent potential temperature, 
  saturated equivalent potential temperature, convective instability, conditional 
  instability, principal component analysis, linear discriminant analysis.<br>
</p>
<p align="justify"><b>1. Introduction</b><br>
  Pre-monsoon thunderstorms over the Eastern Zone of India have many beneficial 
  effects, though in some cases strong wind squall, hail and intense precipitation 
  create havoc. As thunderstorm rain is the major source of water here during 
  the hot pre-monsoon summer days and it gives much relief from hot and humid 
  weather, it is mostly welcome in this region and there is a need to construct 
  a statistical index which may help one to identify an unknown day as a thunderstorm 
  (TS) or non-thunderstorm (NTS) day beforehand. The aim of the present work is 
  to forecast the development /non-development in Calcutta on the basis of 0000 
  UTC and 1200 UTC radiosonde data in Calcutta. The classification rule applied 
  here is based on an index called the &lsquo;Linear Discriminant Function&rsquo; (LDF).</p>
<p align="justify"> The study has been confined up to the 500 hPa level because 
  the importance of this level has already been stressed by a number of scientists, 
  <i>viz</i>. Showalter (1953), Galway (1956), Darkow (1968), Fujita <i>et 
  al.</i> (1970) and Miller (1972). The level of cloud development may also be 
  taken around 500 hPa (Kessler, 1982). </p>
<p align="justify">The work has been initially started with the following 20 thermodynamic 
  and dynamic parameters: (<i>&#952;es</i>-<i>&#952;e</i>), (P-PLCL), <i>&#8706;&#952;e/</i>&#8706;<i>z</i>, 
  <i>&#8706;&#952;es/&#8706;z</i> and <i>&#8706;v</i>/<i>&#8706;z</i> for 
  each of the following atmospheric layers: (1000-850) hPa, (850-700) hPa, (700-600) 
  hPa and (600-500) hPa.</p>
<p align="justify">Here <i>&#952;es</i> and <i>&#952;e</i> are respectively 
  saturated equivalent potential temperature and equivalent potential temperature, 
  P and PLCL denote the pressure at the reference level and that at the corresponding 
  lifting condensation level respectively; <i>v</i> and <i>z</i> are respectively 
  the wind speed in ms<sup><font size="2">-1</font></sup> and geopotential height 
  in meter. </p>
<p align="justify">It is now a well-known fact that thunderstorms are strongly 
  favored by convective instability (determined by <i>&#8706;&#952;e/&#8706;z</i>), 
  abundant moisture at low levels, strong wind shear (measured by <i>&#8706;&#952;v/&#8706;z</i>) 
  and a dynamical lifting mechanism that can release instability (Kessler, 1982). 
  Not only that, the vertical shear of the environmental winds has to match the 
  value of the convective instability for proper development of the convective 
  cloud (Asnani,1992). Williams and Renno (1993) have emphasized conditional instability 
  for supporting electrification and lightning.</p>
<p align="justify">The thermodynamic parameter (<i>&#952;es-&#952;e</i>) was 
  first introduced by Betts (1974). It is considered as the measure of insaturation 
  of the atmosphere for a layer, <i>&#8706;&#952;e/&#8706;z</i> determines the 
  convective stability or instability of a layer. On the other hand, <i>&#8706;&#952;es/&#8706;z</i> 
  decides the conditional stability or instability of a layer and <i>&#8706;v/&#8706;z</i> 
  gives a measure of vertical shear of horizontal wind for a layer. Here the parameters 
  <i>&#952;e </i>and <i>&#952;es</i> have been calculated by the standard 
  formulae introduced by Bolton (1980). PLCL for the surface parcels was considered 
  as the cloud base by Kuo (1965) and hence the parameter (P-PLCL) may be taken 
  as the forcing factor necessary for the saturation of a parcel.</p>
<p align="justify">It may be mentioned that the parameter (<i>&#952;es-&#952;</i>e) 
  has already been examined by Chowdhury <i>et al</i>. (1996) for instantaneous 
  thunderstorm occurrence in Dhaka, Bangladesh and, Ghosh and De (1997) for instantaneous 
  thunderstorm occurrence in Agartala and Ranchi, India. The same parameter has 
  also been studied by Ghosh <i>et al.</i>(1998) before the occurrence of afternoon 
  pre-monsoon thunderstorms at Calcutta, India. In each of the above mentioned 
  studies, the parameter (<i>&#952;es-&#952;e</i>) has been found to play a 
  significant role on thunderstorm days and it also acts as a discriminating factor 
  between TS (thunderstorm) and NTS (non-thunderstorm / fair-weather) days.</p>
<p align="justify"> At first all the 20 parameters have been used to form the 
  discriminating functions for the TS and NTS days. But to deal with all the 20 
  parameters at a time is time-consuming. So, in the second stage of the work 
  the number of the parameters has been reduced to 10 by Principal Component Analysis 
  (PCA). These 10 newly formed parameters have been used to construct the discriminant 
  functions for TS and NTS days. It is very interesting to note that the accuracy 
  of the results has improved significantly in the second stage of the work. The 
  present study has been performed separately for the morning and afternoon as 
  it is clear from a previous work on PCA (Ghosh <i>et al.,</i> 1999) that the 
  weather in the morning differs structurally from that in the afternoon in Calcutta 
  during the pre-monsoon season (March, April and May). </p>
<p align="justify">Here, known TS and NTS days have been grouped separately. Each 
  of these two groups represents a discrete outcome and the forecast consists 
  of a categorical statement that one of these outcomes will occur. The discriminant 
  analysis technique has been used as a tool for forecasting by Miller (1962), 
  Lawson and Cerveny (1985), Ward and Folland (1991) and S&aacute;nchez <i>et al. </i>(1998).</p>
<p align="justify"><b><br>
  2. Data</b><br>
  In the present analysis, daily radiosonde data (viz. geopotential height, pressure, 
  dry-bulb temperature, dew-point temperature and wind-speed) of 0000 UTC and 
  1200 UTC taken in Calcutta for the pre-monsoon months (i. e. March, April and 
  May) of the years 1988, 1990, 1991 and 1993 have been utilized to calculate 
  the above mentioned 20 parameters. To identify the TS and NTS days in Calcutta, 
  Monthly Meteorological Report (M. M. R.) data have been used. It should be mentioned 
  that there are two meteorological observation stations in Calcutta, one in Dumdum 
  (DD) and the other in Alipore (ALP). Whenever either station reports thunderstorm, 
  the day is taken as a TS day for Calcutta. The year 1988 has been treated as 
  &lsquo;unknown&rsquo; to validate the indices which have been constructed using the data 
  of 1990, 1991 and 1993.</p>
<p align="justify"> In the literature, the group X consists of the parameters 
  for NTS days and the group Y consists of the parameters for TS days. It is to 
  be noted that the total number of thunderstorms that occurred during the pre-monsoon 
  season (i. e. March, April and May) in 1990, 1991 and 1993 is 45. So, originally 
  each group includes 45 days and 20 parameters. It has been noticed that during 
  the pre-monsoon season, thunderstorms often occur in Calcutta on consecutive 
  days. That is why, the days without occurrence just before and after the TS 
  days have been chosen as NTS days. The intention is to get a clear idea regarding 
  the behavior of the parameters towards the non-occurrence of the phenomenon, 
  even when the parameters may be favorable for the occurrence of thunderstorms. 
  However, the size of each group has been maintained at 45.</p>
<p align="justify">In the second stage, to form the indices, the linear discriminant 
  analysis has been performed with the two groups, each of which includes the 
  same 45 days as before, but the number of parameters in each group is 10 (new 
  parameters formed by the PCA technique) instead of 20.</p>
<p align="justify"><b><br>
  3. Methodology </b><br>
  Let us suppose that there are <i>k</i> parameters, <i>Z<sub><font size="2">i</font></sub> 
  (i</i> = 1 to <i>k</i>) on which we have the following two sets of observations:</p>
<p align="center"><img src="/img/revistas/atm/v17n1/a01pag4.jpg"> </p>
<p align="justify">In the present study, <i>Zi (i</i> = 1 to 20) denotes the 
  above mentioned 20 parameters; <i>Xij</i> denotes the value of the <i>ith</i> 
  parameter on <i>jt</i>h NTS day; <i>Yij</i> gives the value of the ith parameter 
  on <i>jth</i> TS day. Originally, the work has been started with <i>k</i> 
  = 20 parameters which has been ultimately reduced to <i>M*</i> = 10 by the 
  PCA technique, where <i>M*</i> is the number of newly formed parameters and 
  k is the number of original parameters.</p>
<p align="justify"> In this context, let us discuss the objective of the PCA technique 
  in short. Assuming that there are p variables <i>Xi</i> (i = 1 to <i>p</i>), 
  the following <i>p</i> linear combinations &#958;<sub><font size="2"> <i>i</i></font></sub> 
  (i = 1 to<i> p</i>) wich have been mentioned as newly formed parameters in 
  the text are formed as follows: </p>
<p align="center"><img src="/img/revistas/atm/v17n1/a01pag4b.jpg"></p>
<p align="justify">where <i>Wij</i> denotes the weight of the <i>jth </i>variable 
  for the <i>ith</i> principal componente. The weights are estimated such that:<br>
  i) The first principal component ,&#958;<sub><font size="2"><i>1 </i></font></sub>accounts 
  for the maximum variance in the data, the second principal component, &#958;<sub><font size="2"> 
  2</font></sub>, accounts for the maximum variance that has not been accounted 
  for by the first principal component, and so on.</p>
<p align="center"><img src="/img/revistas/atm/v17n1/a01pag4c.jpg"></p>
<p align="justify">The weights are obtained mathematically by using calculus (Sharma, 
  1996).<br>
  In the first part of the work, Linear Discriminant Analysis (LDA) has been performed 
  with the original k-dimensional raw data vector. But in the second part, the 
  same analysis has been performed with <i>M*</i>-dimensional data vector whose 
  elements are the first <i>M*</i> principal components. Here the PCA has been 
  conducted on the correlation matrix instead of the covariance matrix, since 
  the data vectors are unlike (i. e. they have different units) (Wilks, 1995).</p>
<p align="justify">Since a previous work on PCA with the same 20 parameters (Ghosh 
  <i>et al.,</i> 1999) shows that 93.3 % - 98 % information is covered with 
  the first 10 components which involve the parameters from 1000 hPa to 700 hpa, 
  in the second part of the present work, the upper-level parameters have been 
  neglected. Here <i>m</i> = <i>n</i> = 45, <i>m</i> and <i>n</i> being 
  the number of NTS days and TS days respectively. Each of the groups <i>X</i> 
  and <i>Y </i>has been characterized as follows:</p>
<p align="center"><img src="/img/revistas/atm/v17n1/a01pag5.jpg"></p>
<p align="justify">Without any loss of generality, let us assume that the population 
  underlying each of the groups, X and Y has the same covariance matrix. Then 
  the sample covariance matrices <i>Sx</i> and <i>Sy</i> can be computed from 
  the data matrices as follows:</p>
<p align="center"><img src="/img/revistas/atm/v17n1/a01pag5b.jpg"></p>
<p align="justify">Since the covariance structures are assumed to be equal, the 
  above two matrices are averaged to yield a pooled estimate of the dispersion 
  of the data around their means as follows:</p>
<p align="center"><img src="/img/revistas/atm/v17n1/a01pag5c.jpg"></p>
<p align="justify">where the elements <i>U</i><sub><font size="2"><i>i</i></font></sub> 
  (i = 1 to <i>k</i>) to denote the values of the parameters on a particular 
  day, the nature of which is unknown, the following discriminant functions are 
  calculated to determine which group <i>U</i> belongs to:</p>
<p align="center"><img src="/img/revistas/atm/v17n1/a01pag6.jpg"></p>
<p align="justify">where &lsquo;/ &rsquo; means as usual the transpose of a matrix. <br>
  If &#124;<i>Dx - Du</i>&#124; &gt; |<i>Dy &ndash; Du</i>&#124;, then <i>U</i> belongs 
  to the <i>X</i>-group, i. e., the nature of the unknown day is expected to be 
  as that of an NTS day.<br>
  Similarly, if &#124;<i>Dx &ndash; Du</i>&#124; &gt; &#124;<i>Dy &ndash; Du</i>&#124;, then 
  <i>U</i> belongs to the <i>Y</i>-group, i. e., the nature of the unknown day 
  resembles the nature of a TS day.</p>
<p align="justify"> As mentioned earlier, in the second part of the work the PCA 
  technique has been applied first to reduce the dimensionality of the original 
  data matrix without losing any physical information. Then the LDA technique 
  has been performed in a similar manner, as described above, with the newly formed 
  10-dimensional data matrix. Here also, each of the groups <i>X</i> and <i>Y</i> 
  consists of 45 days. The LDA method has been named PCLDA in this part of the 
  work.</p>
<p align="justify"> Next, for the validation of the results, two types of verification 
  have been performed here, <i>viz.</i><br>
  i) Autoverification;<br>
  ii) Verification with unknown data. <br>
  In &lsquo;Autoverification&rsquo;, the days already involved in the analysis 
  have been treated as unknown observations. Incidentally, the &lsquo;Autoverification&rsquo; 
  (<a href="/img/revistas/atm/v17n1/a01pag9.jpg">Table 2</a>) reveals that the 
  results improve significantly in the second stage of the work (i.e. when the 
  PCLDA technique has been used). So for the final verification with unknown data 
  the PCLDA technique has been applied directly.</p>
<p align="justify"> For the final verification, the data of the year 1988, have 
  been taken as unknown observations which are to be classified. Before verification, 
  the thunderstorms of 1988 have been grouped as morning-TS and afternoon-TS according 
  to the Monthly Meteorological Report (M. M. R.) Here, morning-TS means the thunderstorms 
  occurring between 0000UTC and 1200 UTC and afternoon-TS includes the thunderstorms 
  which occur after 1200 UTC but before 0000 UTC of the next day. Similarly, morning-NTS 
  represents the fair weather between 0000 UTC and 1200 UTC and afternoon-NTS 
  represents that during the next 12 hours.</p>
<p align="justify"><b><br>
  3. Results and verification</b><br>
  In <a href="/img/revistas/atm/v17n1/a01pag7.jpg">Table 1</a>, the results of 
  LDA (with 20 original parameters) and PCLDA (with 10 newly formed parameters), 
  in <a href="/img/revistas/atm/v17n1/a01pag9.jpg">Table 2</a>, the results of 
  &lsquo;Autoverification&rsquo; and in Table 3 and 4, the results of final verification 
  with the data for 1988 have been presented.</p>
<p align="justify"><i>LDA-output (<a href="/img/revistas/atm/v17n1/a01pag7.jpg">Table 
  1</a>)</i><br>
  MDx and MDy respectively denote the discriminant functions for NTS and TS for 
  morning and <i>ADx</i> and <i>ADy</i> denote those for afternoon respectively. 
  The values of the discriminant functions as obtained from LDA are as follows:</p>
<p align="center"><img src="/img/revistas/atm/v17n1/a01pag6b.jpg"></p>
<p align="center"><br>
  <a href="/img/revistas/atm/v17n1/a01pag7.jpg">table 1. Linear discriminant functions for LDA and PCLDA</a></p>
<p align="center"><br>
  <a href="/img/revistas/atm/v17n1/a01pag8.jpg">table 1. Continued</a></p>
<p align="justify"><br>
  Autoverification with the above discriminant functions (<a href="/img/revistas/atm/v17n1/a01pag9.jpg">Table 
  2</a>) reveals that, in the morning, 53.3% TS cases and 46.6% NTS cases are 
  correctly classified and in the afternoon 2.2% TS cases and 44.4% NTS cases 
  are correctly classified.</p>
<p align="justify"> <i>PCLDA-output (<a href="/img/revistas/atm/v17n1/a01pag7.jpg">Table 
  1</a>)</i><br>
  For the morning, the discriminant functions for NTS and TS are represented by 
  <i>PMDx</i> and <i>PMDy</i> and those for the afternoon are denoted by PADx 
  and PADy respectively. The values of the functions as furnished by PCLDA are 
  given below:</p>
<p align="justify"> <i>PMDx</i> = -3.405151, <i>PMDy</i> = -6.592593 and <i>PADx</i> 
  = -7.05927, <i>PADy</i> = 27.83625</p>
<p align="justify"> Autoverification with the above discriminant functions (<a href="/img/revistas/atm/v17n1/a01pag9.jpg">Table 
  2</a>) shows that, in the morning, 64.4% TS cases and 60% NTS cases are correctly 
  classified, whereas, in the afternoon, 93.3% TS cases and 77.8% NTS cases are 
  correctly classified.</p>
<p align="justify"> In <a href="/img/revistas/atm/v17n1/a01pag9b.jpg">Table 3</a> 
  and <a href="/img/revistas/atm/v17n1/a01pag10.jpg">Table 4</a>, only the results 
  of PCLDA for 1988 have been presented. It is to be noted that after categorizing 
  the TS and NTS cases as &lsquo;Morning-TS&rsquo;, &lsquo;Morning-NTS&rsquo;, 
  &lsquo;Afternoon-TS&rsquo; and &lsquo;Afternoon-NTS&rsquo; respectively, the 
  application of PCLDA has interpreted the results in a more meaningful way as 
  follows:<br>
</p>
<p align="center"><a href="/img/revistas/atm/v17n1/a01pag9.jpg">table 2. Autoverification</a></p>
<p align="center">&nbsp;</p>
<p align="center"><a href="/img/revistas/atm/v17n1/a01pag9b.jpg">table 3. Verification of thunderstorm days with new set of data</a></p>
<p align="center"><br>
  <a href="/img/revistas/atm/v17n1/a01pag10.jpg">table 4. Verification of non-thunderstorm days with the new set of data</a></p>
<p align="left"><b><br>
  <br>
  4. Conclusion </b><br>
  The present study reveals that the statistical index constructed above not only 
  helps to classify a TS or an NTS day, but it can also be used for 12-hour forecasts 
  of pre-monsoon weather in Calcutta, India. Moreover, since the nature of the 
  method is objective, it can be expected to produce more accurate results than 
  any other subjective method.</p>
<p align="left"> Hence, it can be concluded that if the LDA technique is to be 
  used for the operational purpose of classifying the TS and NTS days in Calcutta, 
  then instead of dealing with huge data matrices, their dimensionality may be 
  reduced first with the help of the PCA technique without losing any important 
  information.</p>
<p align="left"><br>
  <b>Acknowledgement</b><br>
  The authors thank the India Meteorological Department for the sanction of a 
  research project. The present work forms a part of that project.</p>
<p align="left"><br>
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